3.127 \(\int \frac{x^3}{\sqrt{-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx\)

Optimal. Leaf size=115 \[ -\frac{1}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left (\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right )}{2 \sqrt{2}}+\tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right )-2 \sin ^{-1}(x+2) \]

[Out]

-Sqrt[-3 - 4*x - x^2]/2 - 2*ArcSin[2 + x] + ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x -
x^2])/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/
(2*Sqrt[2]) + ArcTanh[x/Sqrt[-3 - 4*x - x^2]]

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Rubi [A]  time = 0.886921, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433 \[ -\frac{1}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left (\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right )}{2 \sqrt{2}}+\tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right )-2 \sin ^{-1}(x+2) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

-Sqrt[-3 - 4*x - x^2]/2 - 2*ArcSin[2 + x] + ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x -
x^2])/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]]/
(2*Sqrt[2]) + ArcTanh[x/Sqrt[-3 - 4*x - x^2]]

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Rubi in Sympy [A]  time = 115.126, size = 129, normalized size = 1.12 \[ - \frac{\sqrt{- x^{2} - 4 x - 3}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 \left (\frac{x}{3} + 1\right )}{2 \sqrt{- x^{2} - 4 x - 3}} - \frac{1}{2}\right ) \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 \left (\frac{x}{3} + 1\right )}{2 \sqrt{- x^{2} - 4 x - 3}} + \frac{1}{2}\right ) \right )}}{4} - 2 \operatorname{atan}{\left (- \frac{- 2 x - 4}{2 \sqrt{- x^{2} - 4 x - 3}} \right )} + \operatorname{atanh}{\left (\frac{x}{\sqrt{- x^{2} - 4 x - 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)

[Out]

-sqrt(-x**2 - 4*x - 3)/2 - sqrt(2)*atan(sqrt(2)*(3*(x/3 + 1)/(2*sqrt(-x**2 - 4*x
- 3)) - 1/2))/4 - sqrt(2)*atan(sqrt(2)*(3*(x/3 + 1)/(2*sqrt(-x**2 - 4*x - 3)) +
1/2))/4 - 2*atan(-(-2*x - 4)/(2*sqrt(-x**2 - 4*x - 3))) + atanh(x/sqrt(-x**2 -
4*x - 3))

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Mathematica [C]  time = 6.28423, size = 1093, normalized size = 9.5 \[ -2 \sin ^{-1}(x+2)-\frac{i \left (-2 i+5 \sqrt{2}\right ) \tan ^{-1}\left (\frac{66 i \sqrt{2} x^4+40 x^4+54 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+316 i \sqrt{2} x^3+332 x^3+216 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2+469 i \sqrt{2} x^2+920 x^2+297 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x+208 i \sqrt{2} x+964 x+162 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+336}{100 \sqrt{2} x^4+114 i x^4+560 \sqrt{2} x^3+392 i x^3+1004 \sqrt{2} x^2+455 i x^2+736 \sqrt{2} x+284 i x+192 \sqrt{2}+132 i}\right )}{8 \sqrt{1-2 i \sqrt{2}}}+\frac{\left (2 i+5 \sqrt{2}\right ) \tanh ^{-1}\left (\frac{66 \sqrt{2} x^4+40 i x^4+54 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+316 \sqrt{2} x^3+332 i x^3+216 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2+469 \sqrt{2} x^2+920 i x^2+297 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x+208 \sqrt{2} x+964 i x+162 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 \sqrt{2}+336 i}{100 \sqrt{2} x^4-114 i x^4+560 \sqrt{2} x^3-392 i x^3+1004 \sqrt{2} x^2-455 i x^2+736 \sqrt{2} x-284 i x+192 \sqrt{2}-132 i}\right )}{8 \sqrt{1+2 i \sqrt{2}}}+\frac{\left (2 i+5 \sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{16 \sqrt{1+2 i \sqrt{2}}}+\frac{\left (-2 i+5 \sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{16 \sqrt{1-2 i \sqrt{2}}}-\frac{\left (-2 i+5 \sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x+8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}+6 i \sqrt{2}+3\right )\right )}{16 \sqrt{1-2 i \sqrt{2}}}-\frac{\left (2 i+5 \sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (-2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x-8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+3\right )\right )}{16 \sqrt{1+2 i \sqrt{2}}}-\frac{1}{2} \sqrt{-x^2-4 x-3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

-Sqrt[-3 - 4*x - x^2]/2 - 2*ArcSin[2 + x] - ((I/8)*(-2*I + 5*Sqrt[2])*ArcTan[(33
6 - (6*I)*Sqrt[2] + 964*x + (208*I)*Sqrt[2]*x + 920*x^2 + (469*I)*Sqrt[2]*x^2 +
332*x^3 + (316*I)*Sqrt[2]*x^3 + 40*x^4 + (66*I)*Sqrt[2]*x^4 + (162*I)*Sqrt[1 - (
2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] + (297*I)*Sqrt[1 - (2*I)*Sqrt[2]]*x*Sqrt[-3 -
4*x - x^2] + (216*I)*Sqrt[1 - (2*I)*Sqrt[2]]*x^2*Sqrt[-3 - 4*x - x^2] + (54*I)*
Sqrt[1 - (2*I)*Sqrt[2]]*x^3*Sqrt[-3 - 4*x - x^2])/(132*I + 192*Sqrt[2] + (284*I)
*x + 736*Sqrt[2]*x + (455*I)*x^2 + 1004*Sqrt[2]*x^2 + (392*I)*x^3 + 560*Sqrt[2]*
x^3 + (114*I)*x^4 + 100*Sqrt[2]*x^4)])/Sqrt[1 - (2*I)*Sqrt[2]] + ((2*I + 5*Sqrt[
2])*ArcTanh[(336*I - 6*Sqrt[2] + (964*I)*x + 208*Sqrt[2]*x + (920*I)*x^2 + 469*S
qrt[2]*x^2 + (332*I)*x^3 + 316*Sqrt[2]*x^3 + (40*I)*x^4 + 66*Sqrt[2]*x^4 + 162*S
qrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] + 297*Sqrt[1 + (2*I)*Sqrt[2]]*x*Sqrt
[-3 - 4*x - x^2] + 216*Sqrt[1 + (2*I)*Sqrt[2]]*x^2*Sqrt[-3 - 4*x - x^2] + 54*Sqr
t[1 + (2*I)*Sqrt[2]]*x^3*Sqrt[-3 - 4*x - x^2])/(-132*I + 192*Sqrt[2] - (284*I)*x
+ 736*Sqrt[2]*x - (455*I)*x^2 + 1004*Sqrt[2]*x^2 - (392*I)*x^3 + 560*Sqrt[2]*x^
3 - (114*I)*x^4 + 100*Sqrt[2]*x^4)])/(8*Sqrt[1 + (2*I)*Sqrt[2]]) + ((-2*I + 5*Sq
rt[2])*Log[(-2*I + Sqrt[2] - (2*I)*x)^2*(2*I + Sqrt[2] + (2*I)*x)^2])/(16*Sqrt[1
- (2*I)*Sqrt[2]]) + ((2*I + 5*Sqrt[2])*Log[(-2*I + Sqrt[2] - (2*I)*x)^2*(2*I +
Sqrt[2] + (2*I)*x)^2])/(16*Sqrt[1 + (2*I)*Sqrt[2]]) - ((-2*I + 5*Sqrt[2])*Log[(3
+ 4*x + 2*x^2)*(3 + (6*I)*Sqrt[2] + 4*x + (8*I)*Sqrt[2]*x + 2*x^2 + (2*I)*Sqrt[
2]*x^2 - 2*Sqrt[2*(1 - (2*I)*Sqrt[2])]*Sqrt[-3 - 4*x - x^2] - 2*Sqrt[2*(1 - (2*I
)*Sqrt[2])]*x*Sqrt[-3 - 4*x - x^2])])/(16*Sqrt[1 - (2*I)*Sqrt[2]]) - ((2*I + 5*S
qrt[2])*Log[(3 + 4*x + 2*x^2)*(3 - (6*I)*Sqrt[2] + 4*x - (8*I)*Sqrt[2]*x + 2*x^2
- (2*I)*Sqrt[2]*x^2 - 2*Sqrt[2*(1 + (2*I)*Sqrt[2])]*Sqrt[-3 - 4*x - x^2] - 2*Sq
rt[2*(1 + (2*I)*Sqrt[2])]*x*Sqrt[-3 - 4*x - x^2])])/(16*Sqrt[1 + (2*I)*Sqrt[2]])

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Maple [A]  time = 0.019, size = 144, normalized size = 1.3 \[ -2\,\arcsin \left ( 2+x \right ) -{\frac{1}{2}\sqrt{-{x}^{2}-4\,x-3}}+{\frac{\sqrt{3}\sqrt{4}}{24}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{6}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}} \right ) -4\,{\it Artanh} \left ( 3\,{\frac{x}{-3/2-x}{\frac{1}{\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}}}} \right ) \right ){\frac{1}{\sqrt{{1 \left ({{x}^{2} \left ( -{\frac{3}{2}}-x \right ) ^{-2}}-4 \right ) \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-2}}}}} \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x)

[Out]

-2*arcsin(2+x)-1/2*(-x^2-4*x-3)^(1/2)+1/24*3^(1/2)*4^(1/2)*(3*x^2/(-3/2-x)^2-12)
^(1/2)*(2^(1/2)*arctan(1/6*(3*x^2/(-3/2-x)^2-12)^(1/2)*2^(1/2))-4*arctanh(3*x/(-
3/2-x)/(3*x^2/(-3/2-x)^2-12)^(1/2)))/((x^2/(-3/2-x)^2-4)/(1+x/(-3/2-x))^2)^(1/2)
/(1+x/(-3/2-x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt{-x^{2} - 4 \, x - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="maxima")

[Out]

integrate(x^3/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)), x)

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Fricas [A]  time = 0.299971, size = 236, normalized size = 2.05 \[ -\frac{1}{8} \, \sqrt{2}{\left (8 \, \sqrt{2} \arctan \left (\frac{x + 2}{\sqrt{-x^{2} - 4 \, x - 3}}\right ) + \sqrt{2} \log \left (-\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) - \sqrt{2} \log \left (\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) + 2 \, \sqrt{2} \sqrt{-x^{2} - 4 \, x - 3} - \arctan \left (\frac{\sqrt{2} x + 3 \, \sqrt{2} \sqrt{-x^{2} - 4 \, x - 3}}{2 \,{\left (2 \, x + 3\right )}}\right ) - \arctan \left (-\frac{\sqrt{2} x - 3 \, \sqrt{2} \sqrt{-x^{2} - 4 \, x - 3}}{2 \,{\left (2 \, x + 3\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="fricas")

[Out]

-1/8*sqrt(2)*(8*sqrt(2)*arctan((x + 2)/sqrt(-x^2 - 4*x - 3)) + sqrt(2)*log(-(2*s
qrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) - sqrt(2)*log((2*sqrt(-x^2 - 4*x - 3)*x -
4*x - 3)/x^2) + 2*sqrt(2)*sqrt(-x^2 - 4*x - 3) - arctan(1/2*(sqrt(2)*x + 3*sqrt(
2)*sqrt(-x^2 - 4*x - 3))/(2*x + 3)) - arctan(-1/2*(sqrt(2)*x - 3*sqrt(2)*sqrt(-x
^2 - 4*x - 3))/(2*x + 3)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)

[Out]

Integral(x**3/(sqrt(-(x + 1)*(x + 3))*(2*x**2 + 4*x + 3)), x)

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GIAC/XCAS [A]  time = 0.275037, size = 250, normalized size = 2.17 \[ \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{\sqrt{-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac{1}{2} \, \sqrt{-x^{2} - 4 \, x - 3} - 2 \, \arcsin \left (x + 2\right ) + \frac{1}{2} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac{1}{2} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="giac")

[Out]

1/4*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 1/4
*sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) - 1/2*sqrt
(-x^2 - 4*x - 3) - 2*arcsin(x + 2) + 1/2*ln(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2)
+ 3*(sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) - 1/2*ln(2*(sqrt(-x^2 - 4*x - 3
) - 1)/(x + 2) + (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 3)